# Śleszyński–Pringsheim theorem

In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński[1] and Alfred Pringsheim[2] in the late 19th century.[3]

It states that if an, bn, for n = 1, 2, 3, ... are real numbers and |bn| ≥ |an| + 1 for all n, then

$\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+ \ddots}}}$

converges absolutely to a number ƒ satisfying 0 < |ƒ| < 1,[4] meaning that the series

$f = \sum_n \left\{ \frac{A_n}{B_n} - \frac{A_{n-1}}{B_{n-1}}\right\},$

where An / Bn are the convergents of the continued fraction, converges absolutely.